Azimuthal anisotropy of direct photons
aa r X i v : . [ h e p - ph ] J a n Azimuthal anisotropy of direct photons
B.Z. Kopeliovich,
1, 2
H.J. Pirner, A.H. Rezaeian , and Iv´an Schmidt Departamento de F´ısica y Centro de Estudios Subat´omicos,Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile Joint Institute for Nuclear Research, Dubna, Russia Institute for Theoretical Physics, University of Heidelberg,Philosophenweg 19, D-69120 Heidelberg, Germany (Dated: November 30, 2018)The electromagnetic bremsstrahlung produced by a quark interacting with nucleons or nuclei isazimuthally asymmetric. In the light-cone dipole approach this effect is related to the orientation de-pendent dipole cross section. Such a radiation anisotropy is expected to contribute to the azimuthalasymmetry of direct photons in pA and AA collisions, as well as in DIS and in the production ofdileptons. PACS numbers: 13.85.QK,24.85.+p,13.60.Hb,13.85.Lg
I. INTRODUCTION
Direct photons, i.e. photons not from hadronic decay,are of particular interest, since they do not participatein the strong interaction and therefore carry undisturbedinformation about the dynamics of the primary hard col-lision.Here we present the basic color-dipole formalism forcalculating the azimuthal distribution of direct photonsradiated by a quark interacting either with a nucleonor nuclear targets. For this purpose we further developthe dipole approach proposed in [1, 2] for electromagneticbremsstrahlung by a quark interacting with nucleons andnuclei. This technique can be applied to dilepton [3, 4, 5]and prompt photon [6, 7] production in pp , pA and heavyion collisions. It can be also used for calculating theazimuthal angle dependence in the radiation of dileptons,or in deep inelastic scattering.An azimuthal asymmetry appears due to dependenceof the interaction of a dipole on its orientation. Indeed,a colorless ¯ qq dipole is able to interact only due to thedifference between the impact parameters of q and ¯ q rel-ative to the scattering center. If ~b is the impact param-eter of the center of gravity of the dipole, and ~r is thetransverse separation of the q and ¯ q , then the dipole in-teraction should vanish if ~r ⊥ ~b , but should be maximalif ~r k ~b . One can see this on a simple example of dipoleinteracting with a quark in Born approximation. Thepartial elastic amplitude reads,Im f q ¯ qq ( ~b, ~r ) = 29 π Z d q d q ′ α s ( q ) α s ( q ′ )( q + µ )( q ′ + µ ) × h e i~q · ( ~b + ~r/ − e i~q · ( ~b − ~r/ i h e i~q ′ · ( ~b + ~r/ − e i~q ′ · ( ~b − ~r/ i , (1)Here we assume for the sake of simplicity that q and ¯ q have equal longitudinal momenta, i.e. they are equallydistant from the dipole center of gravity. The general caseof unequal sharing of the dipole momentum is consideredlater in (23). We introduced in (1) an effective gluon mass µ which takes into account confinement and otherpossible nonperturbative effects.Integrating in (1) with a fixed α s we arrive at,Im f q ¯ qq ( ~b, ~r ) = 8 α s (cid:20) K (cid:18) µ (cid:12)(cid:12)(cid:12)(cid:12) ~b + ~r (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) − K (cid:18) µ (cid:12)(cid:12)(cid:12)(cid:12) ~b − ~r (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:21) , (2)where K ( x ) is the modified Bessel function. This ex-pression explicitly exposes a correlation between ~r and ~b :the two terms cancel each other if ~b · ~r = 0. II. DIRECT PHOTONS: DIPOLEREPRESENTATION
The radiation of direct photons, which in the partonmodel looks like a Compton process gq → γq , in thetarget rest frame should be treated as electromagneticbremsstrahlung by a quark interacting with the target.In the light-cone dipole approach the transverse momen-tum distribution of photon bremsstrahlung by a quarkpropagating interacting with a target t (nucleon, t = N ,or nucleus, t = A ) at impact parameter ~b , can be writtenin the factorized form [2], dσ qT → γX ( b, p, α ) d ( lnα ) d p d b = 1(2 π ) X in,f Z d r d r e i~p · ( ~r − ~r ) × φ ⋆γq ( α, ~r ) φ γq ( α, ~r ) F t ( ~b, α~r , α~r , x ) . (3)Here ~p and α = p + γ /p + q are the transverse andfractional light-cone momenta of the radiated photon, φ γq ( α, ~r ) is the light-cone distribution amplitude for the qγ Fock component with transverse separation ~r , and F t ( ~b, α~r , α~r , x ) is an effective partial amplitude to bediscussed below. The product of the distribution ampli-tudes, summed in (3) over initial and final polarizationsof the quark and photon, reads [2], X in,f φ ⋆γq ( α, ~r ) φ γq ( α, ~r ) = α em π m q α × n α K ( αm q r ) K ( αm q r )+ [1 + (1 − α ) ] ~r .~r r r K ( αm q r ) K ( αm q r ) o . (4)Here m q is the effective quark mass, which is in fact aninfra-red cutoff parameter, and can be adjusted to pho-toproduction data [8], or shadowing [9], and whose valueis m q ≈ . F t ( ~b, α~r , α~r , x ) is a linear combination of ¯ qq dipole par-tial amplitudes at impact parameter b , F t ( ~b, α~r , α~r , x ) = Im h f tq ¯ q ( ~b, α~r , x ) + f tq ¯ q ( ~b, α~r , x ) − f tq ¯ q ( ~b, α ( ~r − ~r ) , x ) i , (5)where x is Bjorken variable of the target gluons. III. AZIMUTHAL ASYMMETRY INQUARK-NUCLEON COLLISIONS
In the case of a nucleon target ( t = N ), the partialelastic amplitude f N ¯ qq ( ~b, ~r ) of interaction of the ¯ qq dipolewith a proton at impact parameter ~b , is related to thedipole cross section as, σ N ¯ qq ( r ) = 2 Z d b Im f N ¯ qq ( ~b, ~r ) . (6)where σ Nq ¯ q ( r ) is the total cross section of a ¯ qq - protoncollision. Here and further on, unless specified otherwise,the dipole cross section and partial amplitudes implicitlydepend on the Bjorken variable x of the target gluons.The cross section σ N ¯ qq ( r ) has been rather well deter-mined by data on deep-inelastic scattering [10]. Withthis input, and using Eq. (3), one can calculate the in-clusive differential cross section of direct photon emission.This was done in [7] for pp collisions, with results in goodagreement with data.Using the partial elastic amplitude f N ¯ qq ( ~b, ~r ) one canalso calculate the differential elastic cross section ofdipole-nucleon scattering. Neglecting the real part, theamplitude reads, dσ (¯ qq ) Nel ( r ) dq T = 14 π (cid:12)(cid:12)(cid:12)(cid:12)Z d b e i~q T · ~b Im f N ¯ qq ( ~b, ~r ) (cid:12)(cid:12)(cid:12)(cid:12) ≈ [ σ N ¯ qq ( r )] π exp h − B (¯ qq ) Nel ( r ) q T i . (7)In the second line of this equation we rely on the small- q T approximation. This defines the forward slope of the differential cross section, which can be calculated as, B (¯ qq ) Nel ( r ) = 12 (cid:10) s (cid:11) = 1 σ N ¯ qq ( r ) Z d s s Im f N ¯ qq ( ~s, ~r ) . (8)The slope for small-dipole-proton elastic scattering wasmeasured in diffractive electroproduction of ρ -mesons athigh Q at HERA [11]. The measured slope, B (¯ qq ) Nel ( r ) ≈ − , agrees with the expected value B (¯ qq ) Nel ( r ) ≈ B ppel / ~b , and therefore we should tracea correlation between the vectors ~p and ~b . The popularcorrelation function is defined as, v qN ( b, p, α ) = h ˆ v i φ p = 2 * ~p · ~bpb ! + φ p − R π dφ p ˆ v dσ qT → γX ( b,p,α ) d ( lnα ) d p d b R π dφ p dσ qT → γX ( b,p,α ) d ( lnα ) d p d b , (9)where the averaging is performed integrating in (3) overthe azimuthal angle φ p of the transverse momentum ~p . IV. RADIATION PRODUCED BY A QUARKPROPAGATING THROUGH A NUCLEUS
In this case the partial amplitude to be used in (5), fora ¯ qq dipole colliding with a nucleus at impact parameter b , reads,Im f Aq ¯ q ( ~b, ~r ) = 1 − (cid:20) − A σ Nq ¯ q ( r ) ˜ T A ( ~b, ~r ) (cid:21) A ≈ − exp (cid:20) − σ Nq ¯ q ( r ) ˜ T A ( ~b, ~r ) (cid:21) . (10)The effective nuclear thickness ˜ T A is defined as [12],˜ T A ( ~b, ~r ) = 2 σ N ¯ qq ( r ) Z d s Im f N ¯ qq ( ~s, ~r ) T A ( ~b + ~s ) , (11)where the nuclear thickness function is defined as an in-tegral of the nuclear density along the particle trajectory, T A ( b ) = R ∞−∞ dzρ A ( b, z ).Calculating v qA ( b, p, α ), we can average over φ p , * ~p · ~bpb ! + φ p ∝ π Z dφ p ~p · ~bpb ! dσ qA → γX ( b, p, α ) d ( lnα ) d p d b , (12)analytically. Instead of integration over direction of ~p atfixed ~b , one can integrate over direction of ~b at fixed ~p .The advantage of such a replacement is obvious: all the b -dependence in (3) is located in the effective amplitude F A and it has an explicit and simple form.Indeed, the mean value of s is according to (8) h s i ≈ . , which is much smaller than the heavy nucleusradius squared, R A . Therefore we can expand T A ( ~b + ~s )as T A ( ~b + ~s ) = T A ( b ) + ~s · ~bb T ′ A ( b ) + 12 ~s · ~bb ! T ′′ A ( b ) + ... (13)Correspondingly, the partial amplitude (10) can be ex-panded as,Im f Aq ¯ q ( ~b, ~r ) ≈ − exp (cid:20) − σ Nq ¯ q ( r ) T A ( b ) (cid:21) × ( − b T ′ A ( b ) γ ( ~b, ~r ) − b h T ′′ A ( b ) γ ( ~b, ~r ) − T ′ A ( b ) γ ( ~b, ~r ) i) , (14)where γ n ( ~b, ~r ) = Z d s Im f Nq ¯ q ( ~s, ~r )( ~s · ~b ) n . (15)Integrating the amplitude (14) together with ˆ v over φ b we find that the first two terms in the curly bracketsin (14) give zero, and the rest is,Im ˜ f Aq ¯ q ( b, ~r ) ≡ π π Z dφ b Im f Aq ¯ q ( ~b, ~r ) ˆ v ( φ b )= e − σ Nq ¯ q ( r ) T A ( b ) h T ′′ A ( b ) g ( r ) − T ′ A ( b ) h ( r ) i . (16)Here g ( r ) = Z d s Im f N ¯ qq ( ~s, ~r ) (cid:20) ~p · ~s ) p − s (cid:21) ; (17) h ( r ) = Z d s d s Im f N ¯ qq ( ~s , ~r )Im f N ¯ qq ( ~s , ~r ) × (cid:20) ~s · ~p )( ~s · ~p ) p − ( ~s · ~s ) (cid:21) . (18)Eq. (16) shows that the azimuthal asymmetry isstrongly enhanced at the nuclear periphery. Indeed, atsmall impact parameters the amplitude Eq. (16) is sup-pressed by the factor exp[ − σ Nq ¯ q ( r ) T A ( b )], and moreover, T ′ A ( b ) ≈ − ρ b/ p R A − b and T ′′ A ( b ) ≈ ρ R A / ( R A − b ) / are vanishingly small and peak at the periphery( ρ ≈ .
16 fm − is the central nuclear density). Thesmallness of both the amplitude and azimuthal asym-metry justifies also the expansion made in (14). As far as the partial amplitudes f N ¯ qq ( ~s, ~r ) and theirasymmetric part Eq. (16) are known, one can calculatethe azimuthal asymmetry of photons radiated in quark-nucleus collisions, v qA ( b, p, α ) dσ qA → γX ( b, p, α ) d ( lnα ) d p d b = X in,f Z d r d r × e i~p · ( ~r − ~r ) φ ⋆γq ( α, ~r ) φ γq ( α, ~r ) ˜ F A ( b, α~r , α~r ) . (19)where˜ F A ( b, α~r , α~r ) = Im ˜ f Aq ¯ q ( b, αr ) + Im ˜ f Aq ¯ q ( b, αr ) − Im ˜ f Aq ¯ q (cid:0) b, α | ~r − ~r | (cid:1) . (20)Notice that the cross section in the left hand sideof Eq. (19) can also be calculated without using theexpansion (13), relying on the eikonal approximation,˜ T A ( ~b, ~r ) ≈ T A ( b ), which is known to be quite accuratefor heavy nuclei. V. PARTIAL DIPOLE AMPLITUDE f N ¯ qq ( ~b, ~r ) The next step is to model the partial dipole ampli-tude f N ¯ qq ( ~b, ~r ). An azimuthal asymmetry can only emergeif the amplitude f N ¯ qq ( ~b, ~r ) contains a correlation betweenthe vectors ~b and ~r . If such a correlation is lacking, thefunctions Eqs. (17) and (18) are equal to zero. A modelfor f N ¯ qq ( ~b, ~r ) having no ~b − ~r correlation was proposed in[13].It is rather straightforward to calculate the partial am-plitude within the two gluon exchange model [14],Im f N ¯ qq ( ~b, ~r ) = 23 π Z d q d q ′ α s ( q ) α s ( q ′ )( q + µ )( q ′ + µ ) × e i~b · ( ~q − ~q ′ ) (cid:16) − e i~q · ~r (cid:17)(cid:16) − e − i~q ′ · ~r (cid:17) × h F N ( ~q − ~q ′ ) − F (2 q ) N ( ~q, ~q ′ ) i , (21)where F N ( k ) = h Ψ N | exp( i~k · ~ρ ) | Ψ N i is the nucleon formfactor, and F (2 q ) N ( ~q, ~q ′ ) = h Ψ N | exp[ i~q · ~ρ − i~q ′ · ~ρ ] | Ψ N i is the so called two-quark nucleon form factor. Both canbe calculated using the three valence quark nucleon wavefunction Ψ N ( ~ρ , ~ρ , ~ρ ).An effective gluon mass µ is introduced in (21) in orderto imitate confinement. We fix its value at µ = m π inorder to reproduce the large hadronic cross sections.The Born amplitude is unrealistic since leads to anenergy independent dipole cross section σ ¯ qq ( r, x ). Thisdipole cross section has been well probed by measure-ments of the proton structure function at small Bjorken x at HERA, and was found to rise towards small x , withan x dependent steepness. In fact, it can be expressedvia the unintegrated gluon density F ( x, q ), σ ( r, x ) = 4 π Z d qq (cid:0) − e − i~q · ~r (cid:1) α s ( q ) F ( x, q ) . (22)Analogously, the partial amplitude for dipole-nucleonelastic scattering at impact parameter ~b between the cen-ters of gravity of the dipole and nucleon reads,Im f N ¯ qq ( ~b, ~r, β ) = 112 π Z d q d q ′ q q ′ α s F ( x, ~q, ~q ′ ) e i~b · ( ~q − ~q ′ ) × (cid:16) e − i~q · ~rβ − e i~q · ~r (1 − β ) (cid:17) (cid:16) e i~q ′ · ~rβ − e − i~q ′ · ~r (1 − β ) (cid:17) . (23)Here the dipole has transverse separation ~r , fractionallight-cone momenta of the quark and antiquark, 1 − β and β respectively. Since the radiated photon takes awayfraction α of the quark momentum, the correspondingdipole has β = 1 / (2 − α ). The impact parameter ~b ofthe dipole is the transverse distance from the target tothe dipole center of gravity, which is shifted towards thefastest q or ¯ q in accordance with (23).In (23) α s = p α s ( q ) α s ( q ′ ), and we introduced theoff-diagonal unintegrated gluon density F ( x, ~q, ~q ′ ), whichin the Born approximation limit takes the form, F ( x, ~q, ~q ′ ) ⇒ F Born ( ~q, ~q ′ )= 4 α s π h F N ( ~q − ~q ′ ) − F (2 q ) N ( ~q, ~q ′ ) i . (24) Besides, the partial elastic amplitude Eq. (23), shouldsatisfy the conditions Eqs. (6) and (8). For the dipolecross section we rely on the popular saturated shape [10]fitted to HERA data for F p ( x, Q ) and we choose thefollowing form of F ( x, ~q, ~q ′ ), F ( x, ~q, ~q ′ ) = 3 σ π α s q q ′ R ( x ) × exp h − R ( x ) ( q + q ′ ) i × exp (cid:2) − R N ( ~q − ~q ′ ) / (cid:3) , (25)where σ = 23 .
03 mb, R ( x ) = 0 . × ( x/x ) . with x = 3 . × − [10] and x = p/ √ s [7]. Weassume here that the Pomeron-proton form factor hasthe Gaussian form, F p IP ( k T ) = exp( − k T R N / pp elastic differential cross section is B ppel =2 R N +2 α ′ IP ln( s/s ), where α ′ IP ≈ .
25 GeV − is the slopeof the Pomeron trajectory, s = 1 GeV . R N ≈ h r ch i / f N ¯ qq ( ~b, ~r, x, β ) = σ πB el ( exp " − [ ~b + ~r (1 − β )] B el + exp " − ( ~b − ~rβ ) B el − " − r R − [ ~b + (1 / − β ) ~r ] B el , (26)where B el ( x ) = R N + R ( x ) /
8. This amplitude satisfiesthe conditions Eqs. (6) and (8). This expression also goesbeyond the usual assumption that the dipole cross sectionis independent of the light-cone momentum sharing β .The partial amplitude Eq. (26) does depend on β , butthis dependence disappears after integration over impactparameter ~b . VI. NUMERICAL RESULTS
Now we are in a position to calculate v qN ( b, p, α ). Ex-amples of quark-nucleon collisions radiating a photon,with α = 1 and at different impact parameters and en-ergies, are depicted in Fig. 1. The results show thatthe anisotropy of the dipole interaction rises with im-pact parameter, reaching rather large values. As func-tion of the transverse momentum of the radiated pho-tons, v qN ( b, p, α ) vanishes at large p T . Such a behaviorcould be anticipated, since the interaction of vanishinglysmall dipoles responsible for large p is not sensitive tothe dipole orientation.The next step is calculating the azimuthal asymmetryin quark-nucleus collisions. The results are plotted inFig. 2 as function of transverse momentum, at different impact parameters and at the energies of RHIC and LHC.The first observation is the smallness of v qA , whichis suppressed an order of magnitude compared to v qN .At first glance this might look strange, since the quarkinteracts with nucleons anyway. However, a quark prop-agating through a nucleus interacts with different nucle-ons located at different azimuthal angles relative to thequark trajectory. Their contributions to v qA tend to can-cel each other, restoring the azimuthal symmetry. Suchcancellation would be exact if the nuclear profile function T A ( b ) were constant. We have a nonzero, but small v qA only due to the variation of T A with b , i.e. the presenceof finite first and second derivatives, as was derived inEq. (16).The results of a numerical integration (without expan-sion (14)), depicted in Figs. 1-2, also confirm the antici-pation based on Eq. (16) that the azimuthal asymmetryis enhanced on the nuclear periphery.We used the Woods-Saxon parametrization for nucleardensity [15]. The anisotropy of electromagnetic radia-tion appears only on the nuclear periphery and accord-ing to (16) is extremely sensitive to the behavior of thenuclear thickness function at the very edge of the nu-cleus. Electron scattering data, which is the main source T (GeV)-0.2-0.100.10.20.30.4 v N Solid: √ s = 200 GeVDashed: √ s = 5.5 TeV b = 1 fmb = 0.6 fmb = 0.4 fmb = 0.2 fm FIG. 1: The anisotropy parameter v qN ( b, p, α ) as function of p calculated at α = 1 for different impact parameters b andenergies: √ s = 200 GeV (solid, b = 0 . , . , . , √ s = 5500 GeV (dashed, b = 0 . , p T (GeV) -0.06-0.04-0.0200.020.04 v A b=9.5 fmb=6 fmb=7 fmb=8 fm Solid: √ s = 200 GeVDashed: √ s = 5.5 TeV b=5 fm FIG. 2: Azimuthal anisotropy of direct photons with α = 1from quark-lead collisions at different impact parameters asis labeled in the plot. Solid and dashed curves correspondto the energies of RHIC ( b = 5 , , , , . b = 6 , of information about the electric charge distribution innuclei, is not sensitive to the neutron distribution, whichis known to be enlarged on the periphery. Thereforethe details of the shape of the density distribution onthe nuclear surface are poorly known. As a simple esti-mate of the theoretical uncertainty related to this prob-lem one can use an alternative parametrization of the nu-clear density, such as the simple and popular hard sphereform, ρ ( r ) = ρ Θ( R A − r ). We compare in Fig. 3 theanisotropy parameters v qA ( p, b, α ) calculated with hard T (GeV)-0.1-0.0500.050.10.15 v A HS ProfileWS Profile b = 7 fmb = 5 fm
FIG. 3: Azimuthal anisotropy of direct photons with α = 1from quark-lead collisions at b = 5 and 7 fm. Solid anddashed curves are calculated with Woods-Saxon (WS) andhard sphere (HS) parametrizations of nuclear density respec-tively. sphere (dashed) and Woods-Saxon (solid) parametriza-tions. As one could expect, the hard sphere density leadsto a quite larger anisotropy, since the derivatives of thenuclear profile function are much sharper. VII. SUMMARY
Summarizing, we extended the dipole description ofelectromagnetic radiation [1, 2] in quark nucleon and nu-cleus collisions to calculation of the azimuthal angle dis-tribution. This problem involves more detailed featuresof the dipole amplitude, namely its dependence on dipolesize and impact parameter, as well as on their correlation.We propose a simple model generalizing the unintegratedgluon density fitted to HERA data for the proton struc-ture function to an off-diagonal gluon distribution. Thelatter satisfies all the imposed boundary conditions.The developed theoretical tools can be applied to thecalculation of the azimuthal asymmetry in DIS and inDrell-Yan reactions on a proton, as well as to the pro-duction of direct photons and Drell-Yan pairs in proton-nucleus and heavy ion collisions.
Acknowledgments
This work was supported in part by Fondecyt (Chile)grants 1070517 and 1050589 and by DFG (Germany)grant PI182/3-1. AHR is grateful to the hospitality ofHans Pirners group at Heidelberg University where thiswork was started, and acknowledges the financial supportfrom the Alexander von Humboldt foundation. [1] B. Z. Kopeliovich, proc. of the workshop Hirschegg’95: Dynamical Properties of Hadrons in Nuclear Mat-ter, Hirschegg January 16-21, 1995, ed. by H. Feld-meyer and W. N¨orenberg, Darmstadt, 1995, p. 102(hep-ph/9609385).[2] B.Z. Kopeliovich, A. Schaefer and A.V. Tarasov, Phys.Rev.
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